Generalized Surgery on Riemannian Manifolds of Positive Ricci Curvature

TL;DR

This paper generalizes Wraith's surgery theorem for positive Ricci curvature, introducing a new method involving sphere bundles and core metrics to construct positive Ricci curvature metrics on complex manifolds.

Contribution

It extends the surgery theorem by using sphere bundles and core metrics, enabling the construction of positive Ricci curvature metrics on new classes of manifolds.

Findings

Constructed core metrics on 2-sphere bundles with base admitting a core metric

Produced new examples of 6-manifolds with positive Ricci curvature

Generalized surgery techniques for positive Ricci curvature preservation

Abstract

The surgery theorem of Wraith states that positive Ricci curvature is preserved under surgery if certain metric and dimensional conditions are satisfied. We generalize this theorem as follows: Instead of attaching a product of a sphere and a disc, we glue a sphere bundle over a manifold with a so-called core metric, a type of metric which was recently introduced by Burdick to construct metrics of positive Ricci curvature on connected sums. As applications we construct core metrics on 2-sphere bundles, where the base admits a core metric, and obtain new examples of 6-manifolds with metrics of positive Ricci curvature.